.
-
.
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:-.' . .
• . .
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....
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by
D,..J';'id :B. Dime.m
UniversHi3s
0:
0
Florida and l\T')y.th C loUn.:
,e
......
/,{,{~,
'!,:-
This rese~rch was 6u~~orted jointly by th8
Florida Agricultur~l ~xp~riment St~tion, by
thoU. S. Public Heulth Serviea and by the
United States Air Force through the Office
of Scientific Research of the Air Research
and Development Comm~nd.
Institute of Statistics
Mimeogra,h Series No. 161
December, 1956
....
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k't·
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MULTIPLE RANGE TESTS FOR CORRELATED AND RETEROSCEDASTIC MEANS. *
by
I ...
Da,vid B. Dunoan
Universities of Florida and North Carolina
1.
INTRODUCTION
MUltiple range tests have been developed by several writers, for example,
D. Newman
[8]. M. Keuls [5
J,
J. W. Tukey [10] and D. B. Dunoan
[3
J,
for
testing differenoes between several treatment means in oases in which all suoh
differenoes are of equal
~
priori interest. These tests, which are also described
in reoent textbooks, for example, W. T. Federer
[4, cha.pter 2 J, ha.ve been worked
out for data in whioh the treatment means are homoeoedastic
and are uncorrelated.
(ha~e
equal variances)
Reoently, C. Y. Kramer [6] has presented a simple method for
extending these procedures to give useful tests for differences between means with
unequal replioa.tiona, the method being applicable to any set of heteroscedastic unoorrelated means.
In a. subsequent pa,per
[8 J, the same author has given further
extensions to tests of means whioh are also oorrelated, suoh as the adjusted means
from analyses of covariance or from inoomplete blook designs.
Similar work ha,s also
been done by E. Bleicher [1 ] and P. G. Sanders [9 ] in extending a multiple F
test to making tests in lattice and rectangular lattioe designs.
One purpose of this paper is to present a, more complete method for these extensions which necessarily sacrifices a little in simplicity but is more powerful,
especially in cases in which the differenoes between the means have appreciably
different variances. Another purpose is to indicate briefly the closeness of the
properties of these complete tests of heteroscedastic and correlated meane to those
of the corresponding tests of homosceda.stic and uncorrelated means.
Incidental to
these ma,in purposes, a short-cut skipping prinoiple, useful in a,pplying multiple
range tests to a large number of treatment means (or totals), is also presented.
*Research Jointly supported by the Florida Agricultural Experiment Station~ by
the U. S. Public Health Service and by the U. S. Air Foroe through the Offioe of
Scientifio Research of the Air Researoh and Development Command.
- :2 -
BASIC BULE FOB COMPLETE TESTS
:2.
Let ml , m:2' .•• , mn represent n normally distributed means such that the
= kija:2
variance of the difference between ea.oh pair can be written v(mi-m )
j
:2
ki j ia known and a
is an expeoted erDor mean square.
Let s
:2
where
with n:2 degrees of
freedom be the usual type of analysis ot v.ar1ance error mean equate estimate for 0 2 • D1
:2 :2
:2
other words, n:2 s /a is distributed as X n:2 and is independent of the means ~, m2 ,
Call a iJ
=
j:27kiJ
the a.djustment fa,ctor for
= aij(mi-mj )
and (mi-m )'
J
adjusted ,difference between the means m and mj , and call
i
B~
= s.zp
the
the critical
value for p means, where zp is the Studentized significant range for p means for n 2
degrees of freedom and for an a-level test.
The proposed complete basic rule for an a-level multiple range test may then
be expressed as follows:
,,justed differenoe
means not
!:El
two
1!: ~
~contained
~ ~
Any subset!!!. p
subset
in
~
~ ~
oontained
1!: ~
~
to exceed the oritioa.l value
homogeneous subset
~
~­
is homogeneous if the largest
~
homogeneous subset
B;.
Any
~
significantly different.
~ ~
signifioantly
different.
This is the same as the basic rule implicitly adopted by Kramer
[6 ]
except
tha,t in the latter a, subset of p means is deolared homogeneous if its a,djusted range
does not exceed
B;.
If an a.djusted difference within a subset exceeds the adjusted
range, a,s it ma,y do through ha,ving a smaller variance and henoe a, larger adjustment
factor, it will be significant by the complete rule and this may also result in the
detection of further significant differences.
3. NOMERj:eAL EX.AM;PLE I: TEST OF UNEQUALLY-REPLICATED MEANS.
Ta,ble 1 illustrates e, convenient method for a.pplying the complete rule.
The
example consists of the applica,tion of a 5 per cent level new multiple range test
[3 ]
to a, set of seven unequally-replica,ted trea,tment means from a. completely ran-
domized design.
A similar extension of any mUltiple range test, e.g' J
[5],
L8 ],
and [10], could be made by the same method, the only difference being in the
source used for the Studentized significant ranges z , in section (b). Table 2
p
gives deta,i1s of the computa,tion of the adjusted differences used in Ta.b1e 1.
The initial prepara.tion of the data, is the same as for Kramer's method
[6
J.
Table 1, section (a) shows the analysis of variance concluding with the calcu1a,tion
of the error standard deviation s
= 'B .45.
Section (b) shows the computa.tion of the
critical values R'p = s.z p , the Studentized significant ranges zp hating been taken
from [3, Table II ] for a, 5 per cent level test entering at the row for n2 = 16
degrees of freedom. Section (c) shows the trea.tment means ranked in ascending order
together with their respective replication numbers in parentheses.
In any test of
uncorrelated means it is helpfUl to list under the ranked means measures, such as
replication numbers in this case, which provide a, quick method of visually assessing
the rela,tive me,gIlitudes of the variances of the means and hence of the variances of
the differences between them.
The me,in part of the test is in the sequences of steps in section (d).
step consists of an application of the ba.sic rule to a, particular subset.
Each
Sequence
1 consists of steps involVing all subsets in which the top mean G is the largest
mean, sequence 2 involves all subsets in which the second mean E is the largest
mean, and in general, sequence i involves all subsets in which the i-th mean is the
largest mean.
The order of steps in each sequence is the same at the beginning a,s that of
previous procedures
[3 ] and [6]. The steps in sequence 1, for example, consist
of testing the adjusted ranges first of the whole set DFABCEG, then of the subset
FABCEG, then of ABCEG and so on. At each step the lowest mean is dropped to give the
subset for the next test.
- 4TABLE 1.
5 Per Cent Level New Multiple Range Test [3 ] of Seven tJilequallyReplicated Means
a.)
Analysis
E!
Variance
Source
Between treatments
-d.f.6
Error
b)
c)
d)
16
p= s
Critioal Values:
p:
(2)
R
.Z
~
= ;m-:s:
r; .45
5,394.6
( 4)
3.15
3.23
zp:
3.00
R' p :
220.4
231.4
237.2
Ranked Trea,tment Means ~ Replioation Numbers
D
680
(3)
Sequenoes
s
p
(3)
(6)
(7)
3.34
245.3
3.37
247.5
F
A
B
C
E
734
(2)
743
(5)
851
( 5)
873
(3)
902
(2)
Seq.
Steps
Imsult
2.
> R'7,(G-F)' > R'6,{G-A)' > R'5,{G-B)' l' R'4
(E-D) , > R'6' (E-F) ':j> R' 5' FABCE: (E-A) '> R' 5' FBCE: (E-F) I >
3,
R'4' BCE,
(C-D)'>R'5,(C-F)'~R'4' FABC:(C-A)':>R'4' FBC:(C-F),
1.
4.
5.
e)
m.e.
(GooD)'
-;j>
R'3' (C-B) ':::f> R '3' (Boo F) , ::j> R' 3 •
(B-D)'>R'4,(B-F)':j>R'3' FAB:(B-A)'>.R'3' FB.
'1> R3 •
(FBe)
(DFA)
(A-D)
Fina.l Result
(FBC)
(DFA)
(BCEG)
AD:y two means not appearing together within the same parentheses are
~ign1ficantly different.
(BCEG)
Any two means
a~pear1ng
parentheses are not signifioantly different.
together within the same
- 5 TABLE 2.
Arithmetical Details for Calou1ating Adjusted Differences in Table 1,
section (d).
= (G-D)a GD =
the replication num"oers for the respective meane, thuB (G-D) ,
265 J2(3) <:5 )/6
(G-F) ,
= 265(1.732)
=~459.0.
Similarly
=
211(1.549)
=
326.8,
(G-A) ,
= 202(1.936) =
391.1,
(0-») , . =
. 94(1.9~6)
=
182.0,
(E-D) ,
=
222(1.549)
=
343.9,
(E-F) , = 168(1.414) = 237.6,
(E-A) ,
=
159(1.690)
268.7,
(O-D) ,
=
334.3,
(O-F) ,
=
139(1.549)
=
=
215.3,
(O-A) ,
= 130(1.936) =
= 117(1.690) =
251.7,
(O-B) ,
=
22(1.936)
=
42.6,
197.7,
(B-D)' = 171(1.936) = 331.1,
(E-F)
I
(B-A) ,
=
193(1.732)
108(2.236)
=
= 241.5,
(A-D) ,
=
63 (1.936)
=
122.0.
The changes in the complete test come in each sequence when the a,djusted
range of a Bubset of p means fa,i1s to exceed R' • If the a,djustment factor for the
p
range is smaller than that of any other difference in the SUbset, or, in a. test like
this of unequally-replicated means, i f either of the extreme means ha.s fewer replica,.
tiona than any of the other means, any of the other differenoes with a. larger a.d.juetment fa.ctor, should aleo be tested.
In such ca,ses it is helpful to write down
the subset concerned a,s a. reminder that it will still be the subset under test until
an adjusted difference within it is found to exceed R'p.
For example, when the ad-
justed range (O-F) , of FABO fa,ils to exoeed R'4 in step 2 of sequence 3, the subset
FABO is written down before testing the
adj~sted
difference (O-A) , in the next step.
This serves a,s e, reminder tha.t (0- A)' must be compared with R' 4 and not R'3 as would
otherwise
ing of
no
ha~e
happened.
Similarly in the next three,steps, the preliminary record-
serves as a reminder tha.t (O-F)', (C-B) , and (E-F)
compared with R' 3 •
I
eaoh ha,ve to be
- 6 -
When an adjusted differenoe between the top mean in a subset of p means and
an internal mean is found to exoeed B' p , the internal mean is dropped to give the
next subset to be tested, For example, in step' of sequence' FABC is reduoed to
FBC in this way by droppifig A when (C-A) , exoeeds B'4.
When an adjustee difference, not involving the top mean, is found to exoeed
B'p' two subsets ma.y qua,l1fy for further testing in the same sequenoe. For example,
if four means were ranked and ha.d replication, numbers as follows
P
S
(1)
(50)
Q
(1)
(100)
the testing steps oould be
(Q- p)
I
:j> B ' 4 .PSBQ : (Q- s)
I
::r
B ' 4' (Q- B) ,
t
R ' 4' (R- p)
I
:r
B I 4' (R- s) ,
't R4'
and the subsets lm.Q and PSQ would both qualify for testing in further steps in the
same sequenoe.
The testing for a subset termina.tes either when it is shown to be homogeneous,
which fa.ot is reoorded by noting the subset in parentheses in the result oolumn a.t the
end of the sequence involved, or when the subset is
fo~~d
to be completely inoluded
within another subset already shown to be homogeneous. For example, BeEf} is reoorded
(BCEG) at the end of the first sequenoe to denote its homogeneity.
the fa.ct tha.t the adjusted range (B-G)
This follows from
of BCEG does not exceed R' 4 and neither of
the replioations 5 and' of Band G is less than the replications' or 2 of C or E.
I
The result (DFA) of sequenoe 5 is of a similar form.
sult (FBC) in sequence "
In other cases, e.g., the re-
it is sometimes necessary to test ea.oh a.djusted difference
in the subset before it can be declared homogeneous.
Sequences 2 and 4 provide no additional homogeneous subsets because they
terminate at subsets BCE and FB, whioh are included in (BCEG) and (F.BC) respectively
already shown to be homogeneous.
It should be noted in oonclusion that it is possible for more than one homogeneous subset to be found in a single sequence.
For example, in the case of the
- 7means PS'RQ discussed above the sequence 'involved could termina.te with the results
(PRQ) and (PSQ) or even (~) (1$Q.) and (R~) depending on the other da,ta. involved,
Section (e) of Table 1 shows a useful way of presenting the results of the
test.
The device of presenting the whole set with homogeneous subsets underscored
as is done in
[3 J
and
[6]
cannot be used here because of the differences in the
order of the means in the various homogeneous subsets.
right of F in (DFA) but not in (FBC).
For example, A is to the
The new method of putting homogeneous groups
in parentheses can also be used in tests of equally replica,ted means Bnd ma.y be preferred for printing purposes.
4 • NUMERICAL EXAMPLE II: TEST OF TREATMENT TOTALS IN A SIMPLE LATTICE DESIGN
(Including the Use of Skipping Short Cuts),
Ta,ble 3 1l1ustra.tes the a,pplication of a similar 5 per cent level test to the
Ta,ble 3.
Section (a.), Table 3, shows the value of s obtained from the error mean square
s2 (denotedE in [2]) for the experiment and its delI"ees of freedom ~, Section
e
(b) shows the adjustment fa.ctors for differences between trea,tment totals (totals
being more convenient than means to use in a case like this) ,
Cochran and Cox give
e:e[l+(n-l)~ J,
(their n being the number (2) of repeti-
tions involved) for the estima,ted variance of a difference between two means for
trea,tments in the same block.
2
Thus, using k++ a to denote the variance of a. dif-
ference between totals for treatments in the same block we have k++
Then using a++ for the corresponding adjustment factor, we have a++
(r [1+ (n-l) ~
J'- (1/2).
= 2r~+(n-l)~J.
= J2/k++ =
Similarly, if a+_ is used to denote the a,dJustment fa,ctor
for differences between totals of trea:tments not in the same block, we ha.ve a.
+J2/k+_ = (r [1 +
n~ Jr' (1/2).
In this example, r = 4, n
a,djustment fa,ctors work out to be as shown in section (b) ,
= 2,
1.1
=
= 0.1270 and the
- 8 ~
Seotion (c) shows the ranked treatment totals and critical values reqUired
for the test.
The arrangement is different from the oorresponding sections of the
previous example solely because of the largeness of the number (25) of treatments involved.
The new arrangement is convenient for applying a, skipping method which short
cuts many of the steps at the begi:cning of each sequence.
procedure is unchanged.
In all other respects the
In the first column the treatments 1 1 2 1
he~e been denoted in [2] are redenoted A, B,
"'1
••• ,
25 as they
Y for convenience in the record-
ing of trea,tment sUbsets. The number (i· j) a:ppearing after each treatment letter
denotes the blooks in whioh the treatment falls.
Thus
(,.1) after trea.tment K shows
that it belongs to block' and to block 1 in the first and second types of replicates I
respect!vely.
These numbers are useful in indica,ting which treatments do and which
do not a,ppear together in the same block and thus which adjustment factor applies to
ea,ch difference.
The seoond column of section (c) shows the adjusted treatment totals from
-_2 ] followed by doubly adjusted treatment totals in parentheses Which, for brevitYl we
will
ca,ll treatment totals and adjusted treatment totals, :,respe"tively.
'
Each adJust-
ad treatment total in parentheses is obtained by multiplying the corresponding treatment total by the smallest adjustment factor.
In this example there are only two ad-
Justment factors, the smaller being .447, hence the adjusted total for K, for instance, is 88.4( .447)
= '9.5
as shown.
The column of adjusted totals is a new feature
needed in the skipping short cut steps.
The last two oolumns of section (c) show the Studentized ranges z and the
p
critical values R Ip = 3.69 zp • The middle column for p helps in identifying the z
p
and HI values. The z values in this example are obtained from [3, Table II ] for
p
p
a 5 per cent level test the same
8S
in the previous example except that for larger
values of p some simple linear interpolation is needed. 1Ilhen a large number of
treatments is involved as in this example, not all of the critical values R I
be
re~uired.
Each one should thus be obtained only as needed in the
p
se~uence
will
steps.
In Table 3, for instance, only 12 of the possible 24 RIp values are ultimately found
to be needed.
- 9 -
Table 3. 5 Per Cent Level Multiple Range Test of Adjusted Treatment Totals from a
~5
a)
~
Analysis
~
Simple Lattice Design.
Variance
n2 = 56,
s2 = 13.60,
s
II
3.69
b) Adjustment Factors for Differences between Treatment Totals
Two treatments in same block:
a++ = .471
Two treatments not in same block: a+- = .447
c) Ranked Treatment Totals and Critical Values
---
Treatment
11 K (3.1)
2 B (1.2)
15 0 (3.5)
14 N (3.4)
24 X (5.4)
22 V (5.2)
1 A (1.1)
21 u (5.1)
4 D (1.4)
16 P (4.1)
23 w (5.3)
25 Y (5.5)
13 M (3.3)
18 R (4.3)
20 T (4.5)
12 L (3.2)
5 E (1.5)
7 G (2.2)
6 F (2.1)
10 J (2.5)
17 Q (4.2)
8 H (2.3)
3 C (1.3)
9 I (2.4)
19 s (4.4)
Total
88,4 (39.5)
77.3 (34.6)
74.7 (33.4)
71,6 (32.0)
70.6 (31,6)
68.1 (30,4)
66.6 (29.8)
61.4 (27.4)
58.8 (26.3)
58.3 (26.1)
55.7 (24.9)
52 ,7 (23.6)
52.7 (23.6)
52.6 (23.5)
51.6 (23.1)
51.1 (22.8)
50.9 (22.8)
47.6 (21.3)
46.9 (21.0)
46.2 (20.7)
46,0 (20.6)
45.2 (20.2)
44.9 (20,1)
38.1 (17.0)
21.5 ( 9.6)
~
25
24
23
22
21
20
19
18
17
16
15
14
13
12
z
R'
....E
3.47
3,47
3.47
3,47
3.47
3.47
3.46
3.45
3.44
3.43
12.80
12.80
12.80
12.80
12.80
12.80
12.77
12.73
12.69
12.66
3.28
3.25
.10
11.99
-E
11
10
9
8
7
6
5
4
3
2
l~
- 10 ~ §equences
d)
-
Seq.
L.
steps
a
39.5 - R~5 = 26.70, 39.5 -.R = '27.4.
a++(K - u)
2.
Result
34.6 - R~4
> RB, a++(K - A) ':i R7.
s
21.80, 34.6 - R~6
(AVXNOBK)
= 21.94.
a++(B - E) ~ Ri6'
3.
33.4 - R~3
= 20.60,
(ELTRMYWPDUAVXNOB)
33.4 - R~9
= 20.63,
33.4 - Ri8 = 20.67.
a++ (0 - J) > R~8' FGEL ... 0: -
4•
5.
I
I
3'2.0 - R22
= 19. 20, 32,0 - R20
CHQJ •• IN: 31.6 - R~l = 18.80.
6.
30.4 - R~O
7.
29.8 - R~9 = 17.03.
8.
27.4 - R~8
,
last 9.6+ R17
(FGELTRMYWPDUAVXNO)
= 19.20.
(CHQJFGELTRMYWPDUAVX{IJ)
= 17.60.
= 14.67,
(ICHQJFGELTRMYWPDU)
,
= 22.29,9.6+
R8 = 21.70,
( SICHQJFG) •
e)
Final Results
(SICHQJFG) (ELTRMYWPDUAVXNOB)
(ICHOJFGELTRMYWPDU) (AVXNOBK)
(CHOJFGELTRMYWPDUAVXN)
(FGELTRMlWPDUAVXNO )
Any two treatments
significantly different.
~
appearing together within the same parentheses are
Any two treatments appearing together within the same
parentheses are not significantly different.
Table 4. Arithmetical Details for Calculating Adjusted Differences in Table 3, section (
(K - U)
,
Similarly
(K - A)'
(0 - J)'
= a++(K
- U)
= .471(88.4
= .471(21.8) = 10.27,
= .471(28.5) = 13.42,
- 61.4}
(B - E)'
(Q - S)'
= .471(27.0} = 12.72
= .471(26.4} = 12.43,
= .471(24.5) = 11.54.
- 11 -
Section (d) shows the main part of the test arranged in steps within sequences
as in the previous example.
The first two steps in sequence 1 are skipping steps and
short cut the individual testing of 17 differences.
In the first step, the largest
critical value, R;5 = 12.80 is subtracted from the largest adjusted total 39.5 (forK)
giving 26.70.
From this it is concluded that all treatments with adjusted totals be-
low 26.70, namely D, P, ••• , S, are significantly lower than K.
They can thus be
dropped from the set leaving the subset UA •••K for testing in the second step.
The truth of this conclusion is readily seen as follows: -- C0nsider anyone of
the differences concluded significant, say K - M for example. We have "
M( .447)
< K( .447) - R~5
implying (.K - M) (.447)
since RI> '
R , and hence (K - M) ,
25
13
> R~5'
thus (K- M) (.471)
> R~3'
> R13 • Similarly each of the adjusted differences
I
concerned exceeds its corresponding critical value and all are thus significant.
In the second step testing the 8.treatment subset tJA ... K' in· a. similar
way,
the
a
largest critical value R involved is subtracted from the top adjusted total giving
27.4. If there were any further adjusted totals below 27.4 the treatments concerned
could be dropped off and another similar step would be applied.
In these data no
further treatments can be dropped and the skipping method terminates at this second
step. The remainder of the sequence is finished by steps of the type already described
in the first example, and for which the arithmetical details are given in Table 4.
Thus in step 3, a++(.K - u)
I
> R8 ,
,
and in step 4, a++(K - A) ":I R • This terminates the
7
sequence since K - A has the larger adjustment factor a++ and no other difference
I
within the subset AV •••K can exceed R • This result may be usefully recorded as be-
7
fore by putting the subset in parentheses as shown in the result column of section
(d) at the end of sequence 1.
Sequence 2 is very similar to sequence 1. The skipping procedure starts by
I
adding R24 to the second highest adjusted total
34.6 for B and terminates again in
the second step. The remainder of the sequence terminates at the third step with
EL •••B being found homogeneous.
- 12 -
In
s~quence
3 an additional treatment is dropped in the second step and the
skipping procedure extends to the third step.
Contmuing the remainder of the sequence,
a++(O - J) is found to exceed R~6 in the fourth step, a+.(O - F) is known not to
1
exceed R from the preceding skipping steps so FGEL ••• O is recorded at the fifth
17
for further internal testing.
The largest (0 - 1) difference with an a++ adjustment
However a++(O - E) cannot exceed R~7 since a++(B - E)
factor is 0 - E,
sequence 2 hence FG ••• O is homogeneous and the sequence terminates.
only two skipping steps and terminates in a similar way.
1 R~6
in
Sequence 4 has
Sequences 5, 6 and 7 each
terminate rapidly at the first step when the subsets concerned are found to fall
entirely within (CH •••N) of sequence
4. In sequence 8, the difference between the
adjusted total for I, that is, 17.0 and the critical level 14.67 is
60
great as to
leave no doubt of the final result (IC,",U) at the end of the first step.
As soon as all treatments but the lowest have been included in homogeneous
sUbsets,
as is the case in the example at the end of sequence 8, the test can be
completed in one reverse type of sequence working-from the bottom total.
The
reasons for this will be evident from the steps of the last sequence in section (d).
The largest possible homogeneous subset in which the bottom treatment S could be
included at this stage is SI •••D and contains 17 treatments.
obtain S + R~7
= 9.6
+ 12.69
= 22.29.
The first step is to
From this it follows that all treatments with
adjusted totals above 22.29, that is E, L, T, ••• are significantly larger than S.
This leaves the 8-treatment subset SI ••.G for testing in the next step.
In the
·1
second step S + R8 = 21.70, no additional adjusted totals exceed this and the skipping
procedure terminates. Since the range G - S of the subset has the adjustment factor
S
a+ _ we already know from the step 2 that a+ _(G - S) 1> R hence SICHOJFG is recorded
~
for internal testing.
The largest (1 - S) difference with the a++ adjustment factor
is (Q - s) and this is therefore tested in step 3.
Since a++(Q - S) ~ R the
subset is homogeneous, and is recorded (SICHOJFG).
This terminates the test.
S
- 13 Section (e) of Table 3 shows theoomplete summary of the test results.
In
this example the ordering of treatments does not vary from one homogeneous subset to
another.
In such a case the method of reilresent1ng the resl,ilts by underseoring a Ri.:n.=
gle set of tveatment: letters as. .1s_.done in
£;.:.7 may. be: 'used
if preferred.
5. NOTES ON THE PROPERTIES OF THE PROPOSED TEST
Two-mean significance levels:
A two-mean significance level in a test ofn
means may be defined £3_7 as the maximum probability of finding a significant
difference between any two means m and mj given that
i
~j
~i
= ~j
where
= E(mi )
~i
and
= E(m j ). This may be written as the max.p£DijJ ~i = ~Lj_7 where Dij denotes the
decision that mi and m are significantly different.
J
= ~j_7 =
In any a-level test of the proposed type we have max.p~Dij\~i
~
p~aijlmi
-
m~ > s'Z2,n2,a\~i = ~j_7.
Since Z2,n ,a.
2
=J2
t n2 ,a. where t n2 ,a the
a-level (two-sided) significant value of t n , a. t statistic with n2 degrees of freedom,
2
and since the variance of mi - mj is 20 /a~jl' this readily reduces to
2
Hence the two-mean significance levels in an a-level test of the proposed type
are exactly a. as desired.
Higher order significance levels
~
power:
In considering these further
aspects of the proposed test it is helpful to study the decision regions for a
5 per cent level test
1'1
= 2,
r2
=3
£3_7 of
three unequally replicated means m , m2 and m with
l
3
and 1'3 = 4 replications and in which n2 =
00
and s
2
=
2
(1
= 1.
If these
- 14 -
%2
Figure 1
5
%
level test} r
l
= 2,r2 = 3,
r
3
= 4,
n2 =
00,
(l = 1.
- 15 -
they are directly comparable to those of the corresponding 5 per cent level test of
three e~ually replicated means (r
l
= r2
= r 3)
shown in Figure 3 of ~3_7. The dis-
tribution function for the points (Xl' x2 ) is the same in both cases, namely a
bivariate normal with variances 2 and 2 and with covariance zero.
Because the strip regions
Widths in each case (2z2
,CD
=2
(~),
(!L-2) and (2,
3) have the same minimum
x 2.77) it follows, as has already been proved, that
the two-mean significance levels are 5 per cent for the test in Figure 1 as well as
for the test with
e~ual
replications.
The only differences between the regions in the two cases is that in Figure 1
0
the angles between the strip regions (~), (~), (~) and (~) are 50 46' ,
670 48' and 610 26' instead of' all being 600 as in the other figure.
(The cosine of
the angle between any two strips (h, i) and (h, j) is given by
The sides of the hexagonal regions (I, 2, 3) are parallel with the corresponding
strips.
Since these hexagons have the same inscribed circle of radius z3
,00
= 2.92
and differ only in haVing a little asymmetry in Figure 1 the three-mean protection
level (the probability P~(xl' x2 )E(1, 2, 3) E(x l ) = E(X2 ) = 0_7 ) for the Figure
1 test is a close approximation to the desired level .9025 obtained in the other
test.
Furthermore, it seems safe to assume that any deviation due to the asymmetry
would be positive.
In terms of three-mean significance levels, the level of the· Figure 1 test
may thus be said to be close to .0975 (= 1 - .9025) as desired and that any deviation
from this appears to be on the negative or conservative side.
The close similarity of the regions of Figure 1 with those of Figure 3
L-3_7
also indicates that the power functions of the Figure 1 test closely approximate the
.~.
desirable ones of the other procedure.
If the means are correlated as well as being heteroscedastic the geometrical
picture is virtually unchanged.
If we let ~cij_7
represent the dispersion
~3
matrix in the case of three means the cosine of the angle between any two stips
(h,i) and (h,j) in a set of regions otherwise stmilar to those of Figure 1 would be
be
2
This may be expressed in terms of the k ij factors (kijO
= c ii
•. 2c ij + c ii )
defined in section 2, as
and the degree of asymmetry involved
depen~on
these.
Similar considerations lead to the conclusion that the higher order levels of
the proposed complete teat and its power functions are reasonably close to the desired
levels and functions existing in corresponding tests of uncorrelated and homoscedastic
means and that the deviations involved appear to be on the conservative side.
- 17 REFERENCES
£1_7 Bleicher, Edwin, "The Extension of the Multiple Comparisons Test to Lattice
Designs," Virginia Polytechnic Institute M.S. thesis, 55 pp., 1954 •
.
.
.'
_r'o
. .... r· ~~ :
Cochran, W. G. and Cox, G. M., "Experimental Des igns," 2nd Ed., John Wiley
and Sons, Inc., New York, 1956.
[)_7 Duncan, David B., "Multiple Range and MUltiple F Tests," Biometrics, 11,
1-42, 1955.
Federer, Walter T., "Experimental Design," Macmillan, New York, 1955.
Keuls, M., "The Use of the 'Studentized Range' in Connection with an Analysis
of Variance," Euphytica, !, 112-122, 1952.
Kramer, Clyde Young, "Extension of Multiple Range Tests to Group Means with
Unequal Numbers of Replications," Biometrics, 12, 307-310, 1956.
Kramer, Clyde Young, "Extension of Multiple Range Tests to Group Correlated
Adjusted Means," Biometrics, 13,
, 1957.
Newman, D., "The Distribution of the Range in Samples from a Normal Population,
Expressed in Terms of an Independent Estimate of Standard Deviation,"
Biometrika, ~, 20-30, 1939.
Sanders, Paul G., "The Extension of the Multiple Comparisons Test to Rectangular Lattices," Virginia Polytechnic Institute M.S. thesiS, 31 pp., 1953.
L10_7Tukey, J. W., "The Problem of MUltiple Comparisons," unpublished dittoed notes,
Princeton University, 396 pp., 1953.